Abstract
It was recently suggested that causal structures are both dynamical, because of general relativity, and indefinite, because of quantum theory. The process matrix formalism furnishes a framework for quantum mechanics on indefinite causal structures, where the order between operations of local laboratories is not definite (e.g., one cannot say whether operation in laboratory occurs before or after operation in laboratory ). Here, we develop a framework for “dynamics of causal structures,” i.e., for transformations of process matrices into process matrices. We show that, under continuous and reversible transformations, the causal order between operations is always preserved. However, the causal order between a subset of operations can be changed under continuous yet nonreversible transformations. An explicit example is that of the quantum switch, where a party in the past affects the causal order of operations of future parties, leading to a transition from a channel from to , via superposition of causal orders, to a channel from to . We generalize our framework to construct a hierarchy of quantum maps based on transformations of process matrices and transformations thereof.
- Received 20 October 2017
- Revised 2 January 2018
DOI:https://doi.org/10.1103/PhysRevX.8.011047
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Whether one event happens before or after another event is a seemingly objective property of the world. Recently, however, the study of causal structures in quantum mechanics has challenged this long-standing assumption, suggesting that causality might be “indefinite.” Researchers would like to know where indefinite causal structures come from, how they can be found, and whether they can be obtained by manipulating situations with definite causal structure. Answers to these questions would shed light on the nature of causality at the quantum-mechanical level. We develop a mathematical framework for exploring these questions and reveal new insights about cause and effect.
We show that, if the dynamics of causal structures is continuous and reversible, the causal structure cannot be changed. Nevertheless, we show that operations performed by a party in the past can continuously (but not reversibly) influence whether the causal order of future events is definite or indefinite.
For a concrete example, imagine that a person, Alice, is in one room and her friend, Bob, is in another. Alice can causally influence Bob via some wire that connects them. To change their causal structure, one would have to either disconnect and reconnect the wire or completely change the wire. The transformation is therefore either not continuous or not reversible.
Our results demonstrate that under physically reasonable assumptions of continuity and reversibility, a world with definite causal order will never become a world with indefinite causal order and vice versa. This insight opens new perspectives regarding physical realizations of indefinite causal structures.